Problem 7 - College Algebra

Question

For each of the following exercises, find the \(x\)-intercept and the \(y\)-intercept without graphing. Write the coordinates of each intercept. $$3 x-2 y=6$$

Step-by-Step Solution

Answer

x-intercept: (2, 0), y-intercept: (0, -3)

Step 1: Understanding Intercepts

The **x-intercept** of a line is the point where the line crosses the x-axis. At this point, y is zero. The **y-intercept** is the point where the line crosses the y-axis, where x is zero.

Step 2: Find the x-intercept

To find the **x-intercept**, set y = 0 in the equation and solve for x. Substitute y = 0 into the equation:\[ 3x - 2(0) = 6 \]This simplifies to:\[ 3x = 6 \]Divide both sides by 3 to isolate x:\[ x = 2 \]Hence, the **x-intercept** is the point (2, 0).

Step 3: Find the y-intercept

To find the **y-intercept**, set x = 0 in the equation and solve for y. Substitute x = 0 into the equation:\[ 3(0) - 2y = 6 \]This simplifies to:\[ -2y = 6 \]Divide both sides by -2 to isolate y:\[ y = -3 \]Hence, the **y-intercept** is the point (0, -3).

Step 4: Write the Coordinates for Intercepts

The coordinates of the intercepts are: - x-intercept: (2, 0) - y-intercept: (0, -3)

Key Concepts

Understanding the X-Intercept

In coordinate geometry, the x-intercept of a line is the specific location where the line intersects the x-axis. At this particular point, the value of y is always zero. To find the x-intercept from a linear equation, simply set y to zero. Then rearrange and solve the resulting equation for x. Using the example from the exercise above, we set y=0 in the equation \(3x - 2y = 6\), which simplifies to \(3x = 6\). Solving for x, we divide both sides by 3 resulting in \(x = 2\). Thus, the x-intercept is located at the coordinate point (2, 0). Key steps include:

• Set all y-values to zero.
• Solve for x accordingly.
• Write the x-intercept as a coordinate, for instance, \((2, 0)\).

Finding the x-intercept is crucial for understanding the behavior and direction of a linear equation graphically on the coordinate plane.

Exploring the Y-Intercept

The y-intercept in coordinate geometry is the exact point where a line crosses the y-axis. At this point, the value of x is always zero. Identifying the y-intercept for linear equations is straightforward by substituting x with zero in the given formula. Applying this to our example, we take \(3x - 2y = 6\) and substitute x with zero, leading to \(-2y = 6\). Solving for y, we divide both sides by -2, yielding \(y = -3\). This indicates that the y-intercept occurs at the coordinate point (0, -3). The method involves:

• Set x to zero in the equation.
• Solve the resulting expression for y.
• Express the result as a coordinate, in this case \((0, -3)\).

Understanding y-intercepts helps in determining the starting position of a line on a graph, providing insight into the linear equation's nature.

Decoding Linear Equations

Linear equations form a fundamental part of algebra and coordinate geometry, characterized by graphs that are straight lines. Such equations typically appear in the form \(Ax + By = C\), where A, B, and C are constants. In our given example, we have the linear equation \(3x - 2y = 6\). You're often tasked with identifying intercepts, which means setting either x or y to zero to find where the line crosses the respective axis. Linear equations have:

• Constant rate of change, known as slope.
• Intersections with both axes that provide useful solution points.

Grasping linear equations equips students to visualize mathematical relationships on a graph and solve practical problems effectively.

Coordinate Geometry Essentials

Coordinate geometry, also known as analytic geometry, is a branch of geometry where algebraic equations are used to describe geometrical shapes. The fundamental idea is to use coordinates, usually represented as \((x, y)\), to determine specific points on the plane. The exercise involves concepts of coordinate geometry by identifying intercepts where the line meets the axes. This helps graphing the equation \(3x - 2y = 6\) more efficiently. Coordinates like (2, 0) and (0, -3) derived from intercept calculations enable precise plotting on the plane. Coordinate geometry benefits include:

• Visualization of algebraic equations through graphical representations.
• Prediction of location and movement within the plane.
• Application in numerous fields like engineering and physics for problem-solving.

As students delve deeper, coordinate geometry enhances their understanding of spatial relationships and mathematical concepts.